Asymptotic distribution and weak convergence on compact Riemannian manifolds

LetM be a compact Riemannian manifold. The discrepancy of two measures onM can be defined by means of geodesic balls onM. It is shown that a sequence of positive measures converges weakly to an absolutely continuous measure (w.r.t. the volume measure ofM) if and only if the discrepancy converges to 0, and the geodesic balls with vanishingv-measure of the boundary constitute a convergence determining class of sets for weak convergence of a sequence of positive measures tov. Estimates for the distance of probability measures with respect to the Kantorovich metric in terms of the discrepancy are obtained.